A martingale is a process whose conditional drift vanishes relative to the available information. The definition is simple, but it unifies partial sums, conditional expectations, likelihood ratios, sequential estimators, and stochastic integrals. Submartingales and supermartingales allow one-sided drift and are the natural objects for convexity and maximal inequalities.
Martingales, submartingales, and differences
Definition. An adapted process \(M=(M_n,\mathcal F_n)_{n\ge0}\) is a martingale if \(M_n\in L^1\) and \[ \mathbb E[M_{n+1}\mid\mathcal F_n]=M_n \quad\text{almost surely}. \] Replacing equality by \(\ge\) defines a submartingale, and replacing it by \(\le\) defines a supermartingale. A sequence \(D_n\) is a martingale-difference sequence if \(D_n\in L^1\), \(D_n\in\mathcal F_n\), and \(\mathbb E[D_n\mid\mathcal F_{n-1}]=0\).
Proposition. For an integrable adapted process \(M\), the following are equivalent: \[ M\text{ is a martingale}; \qquad \mathbb E[M_m\mid\mathcal F_n]=M_n \quad(0\le n\le m); \] \[ M_n=M_0+\sum_{k=1}^nD_k \quad\text{for a martingale-difference sequence }D_k. \]
Proof. Iterating the one-step condition with the tower property gives \[ \mathbb E[M_m\mid\mathcal F_n] =\mathbb E[\mathbb E[M_m\mid\mathcal F_{m-1}] \mid\mathcal F_n]=\cdots=M_n. \] Conversely take \(m=n+1\). For the increment representation, set \(D_k=M_k-M_{k-1}\). Then \(\mathbb E[D_k\mid\mathcal F_{k-1}]=0\). The reverse implication follows by conditioning the telescoping sum.
Corollary. A martingale has constant mean: \(\mathbb EM_n=\mathbb EM_0\). A submartingale has nondecreasing means, and a supermartingale has nonincreasing means.
Canonical constructions
Example (centered sums). If \(X_n\) are independent and integrable, then \[ M_n=\sum_{k=1}^n(X_k-\mathbb EX_k) \] is a martingale for \(\mathcal F_n=\sigma(X_1,\ldots,X_n)\). Independence can be weakened to \(\mathbb E[X_n\mid\mathcal F_{n-1}]=\mathbb EX_n\).
Example (Doob martingale). If \(Y\in L^1\), then \[ M_n=\mathbb E[Y\mid\mathcal F_n] \] is a martingale by the tower property. It is the successive optimal \(L^1\)-compatible prediction of a fixed terminal quantity as information accumulates.
Example (likelihood-ratio martingale). Suppose \(Q\ll P\) on \(\mathcal F_\infty\), with density \(L=\mathrm{d}Q/\mathrm{d}P\), and define \[ L_n=\mathbb E_P[L\mid\mathcal F_n]. \] Then \(L_n\) is a nonnegative \(P\)-martingale and, for \(A\in\mathcal F_n\), \[ Q(A)=\mathbb E_P[L_n1_A]. \] In dominated sequential models, \(L_n\) is the likelihood ratio based on the first \(n\) observations. This construction underlies change of measure, Bayesian marginal likelihoods, and contiguity of statistical experiments.
Convex functions of martingales
Theorem. If \(M_n\) is a martingale and \(\phi:\mathbb R\to\mathbb R\) is convex with \(\phi(M_n)\in L^1\), then \(\phi(M_n)\) is a submartingale. In particular, \(|M_n|\), \(M_n^+\), and \(|M_n|^p\) for \(p\ge1\) are submartingales when integrable.
Proof. Conditional Jensen gives \[ \mathbb E[\phi(M_{n+1})\mid\mathcal F_n] \ge \phi(\mathbb E[M_{n+1}\mid\mathcal F_n]) =\phi(M_n). \]
Predictable transforms
Theorem (martingale transform). Let \(M\) be a martingale and let \(H_k\) be predictable. If every \(H_k\Delta M_k\) is integrable, then \[ (H\mathbin{\cdot}M)_n :=\sum_{k=1}^nH_k\Delta M_k, \qquad \Delta M_k=M_k-M_{k-1}, \] is a martingale starting from zero.
Proof. Predictability makes \(H_k\) \(\mathcal F_{k-1}\)-measurable, so \[ \mathbb E[H_k\Delta M_k\mid\mathcal F_{k-1}] =H_k\mathbb E[\Delta M_k\mid\mathcal F_{k-1}] =0. \] Therefore the transform has integrable, adapted, conditionally centered increments.
The timing is essential. Choosing \(H_k\) after seeing \(\Delta M_k\), for example \(H_k=\operatorname{sgn}(\Delta M_k)\), would turn the increment into \(|\Delta M_k|\) and create positive drift. Predictability is the mathematical form of “no foresight.”
Doob decomposition
Theorem (discrete Doob decomposition). Every integrable submartingale \(X_n\) has a unique representation \[ X_n=M_n+A_n, \] where \(M\) is a martingale, \(A_0=0\), and \(A_n\) is predictable and increasing.
Proof. Define \[ \Delta A_n =\mathbb E[X_n-X_{n-1}\mid\mathcal F_{n-1}], \qquad A_n=\sum_{k=1}^n\Delta A_k. \] Submartingality gives \(\Delta A_n\ge0\), and \(\Delta A_n\in\mathcal F_{n-1}\), so \(A\) is predictable and increasing. Put \(M_n=X_n-A_n\). Then \[ \mathbb E[\Delta M_n\mid\mathcal F_{n-1}] = \mathbb E[\Delta X_n\mid\mathcal F_{n-1}] -\Delta A_n=0. \] For uniqueness, if \(X=M+A=M'+A'\), then \(B=M-M'=A'-A\) is both a martingale and predictable with \(B_0=0\). Its increment \(\Delta B_n\) is \(\mathcal F_{n-1}\)-measurable and has conditional mean zero, hence \(\Delta B_n=0\) almost surely.
Example. For a square-integrable martingale \(M\), \(M_n^2\) is a submartingale. Its Doob compensator is \[ A_n=\sum_{k=1}^n \mathbb E[(\Delta M_k)^2\mid\mathcal F_{k-1}], \] the predictable quadratic variation developed below. Thus the abstract decomposition directly identifies accumulated conditional variance.